The paper presents general solutions for fractional state-space equations. The analysis of the fractional electrical circuit in the transient state is described by the equation of the state and space equations. The results are presented for the voltage of a capacitor and current in a coil, for different alpha values. The Caputo and conformable fractional derivative definitions have been considered. At the end, the results have been obtained.

An analysis of a given electrical circuit using a fractional derivative. The statespace equation was developed. The dynamics of tensions described by Kirchhoff’s laws equations. The paper used the definition of the integral derivative Caputo and CDF conformable fractional definition. An electrical circuit solution using Caputo and CDF defini- tions for rectangular with zero initial conditions was developed. The results obtained using the Caputo and CDF definitions were compared. The solutions are shown for capacitor voltages, for fractional derivative orders of 0.6, 0.8, 1. The results were compared using graphs.

The use of fractional-order calculus for system modeling is a good alternative to well-known classic integer-order methods, primarily due to the precision with which the modeled object may be mapped. In this study, we created integer and fractional discrete models of a real object – a highspeed brushless micro-motor. The accuracy of the models was verified and compared.

The global stability of positive continuous-time standard and fractional order nonlinear feedback systems is investigated. New sufficient conditions for the global stability of these classes of positive nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple examples of positive nonlinear systems.

The paper discusses the modelling of magnetic coupling in ignition coils by fractional differential equations. The use of fractional-order coupling allows us to consider the losses caused by the non-linearity of the ferromagnetic core of the ignition coil and obtain the waveform of the ignition coil’s secondary voltage closest to the values obtained experimentally.

This work proposes an optimum design and implementation of fractional-order Butterworth filter of order (1 + α), with the help of analog reconfigurable field-programmable analog array (FPAA). The designed filter coefficients are obtained after dual constraint optimization to balance the tradeoffs between magnitude error and stability margin together. The resulting filter ensures better robustness with less sensitivity to parameter variation and minimum least square error (LSE) in magnitude responses, passband and stopband errors as well as a better –3 dB normalized frequency approximation at 1  rad/s and a stability margin. Finally, experimental results have shown both lowpass and highpass fractional step values. The FPAA-configured outputs represent the possibility to implement the real-time fractional filter behavior with close approximation to the theoretical design.

In this paper we discuss the linear quadratic (LQ) optimization problem subject to fractional order irregular singular systems. The aim of this paper is to find the control-state pairs satisfying the dynamic constraint of the form a fractional order irregular singular systems such that the LQ objective functional is minimized. The method of solving is to convert such LQ optimization into the standard fractional LQ optimization problem. Under some particularly conditions we find the solution of the problem under consideration.

The article focuses on the fractional-order backward difference, sum, linear time-invariant equation analysis, and difficulties of the fractional calculus microcontroller implementation with regard to designing a fractional-order proportional integral derivative (FOPID) controller. In opposite to the classic proportional integral derivative (PID), the FOPID controller is defined by five independent parameters. Hence, it is more customizable and, potentially, more precise on condition that the values of fractional integration and differentiation orders are properly selected. However, a number of operations and the time required to calculate the output signal continuously increase. This can be a significant problem considering the limitations of a microcontroller, including memory size and a constant sampling time of the set-up analog-to-digital (ADC) converters. In the article, three solutions are considered, and results obtained in the experiments are presented.

The main goal of introducing Active Suspension System in vehicles is to reduce the vehicle body motion under road obstacles which improves the ride comfort of the passenger. In this paper, the Full Car Model (FCM) with seven Degrees of Freedom is considered and simulated by MATLAB/Simulink. The Terminal Sliding Mode Controller (TSMC) and Fractional Order Terminal Sliding Mode Controller (FOTSMC) are designed to enhance the ride quality, stability and passenger comfort for FCM. The designed FOTSMC has the ability to provide higher control accuracy in a finite time. The performances of the designed controllers are evaluated by measuring the vehicle body vibration in both angular and vertical direction under bump input and ISO-8608 random input against passive suspension system. The FrequencyWeighted Root Mean Square (FWRMS) and Vibration dose value of Body Acceleration as per ISO-2631 are evaluated for FOTSMC, TSMC and PSS. The stability of the FCM is proved by Lyapunouv theory. Further analysis with sprung mass and speed variation of FCM demonstrate the robustness of proposed controller. To investigate the performances of designed controllers, comparison is made with existing Sliding Mode Controller (SMC) which proves that the designed FOTSMC performs better than existing SMC.